Page 352 - DMTH502_LINEAR_ALGEBRA

Linear Algebra

Notes Show that
n n
2
y i = x 2 j
i 1 j 1
2. If M be an n n matrix over C with columns M , M , ... M . Show that M belongs to O(n, c)
1 2 n
if and only if
M j M = .
+
k jk
32.3 Summary

 In this unit certain groups preserving the bilinear forms is studied and seen that these set
of groups is isomorphic to the n × n pseudo orthogonal group when the bilinear form is
non-degenerate.
 The examples of rotation and Lorentz transformations that preserve certain bilinear forms
are studied.
32.4 Keywords

Orthogonal group: The group preserving f given by
n
f( , ) = x y
i i
i 1
for = (x , x , ... x ), = (y , y , ... y ), is called the n-dimensional (real or complex) orthogonal
1 2 n 1 2 n
group.
4
Pseudo-orthogonal Group: For a non-degenerate bilinear form f on R with quadratic form
p n
q(x , x , ... x ) = x 2 j x 2 i
1 2 n
j 1 i p 1
the group of matrices preserving a form of this type is called pseudo-orthogonal group.

32.5 Review Questions

2
1. Let f be the bilinear form on C defined by f[(x , x ), (y , y )] = x y – x y
1 2 1 2 1 2 2 1
show that
2
(a) if T is a linear operator on C , then f(T , T ) = (det T) f( , ) for , in C 2
(b) T preserves f if and only if det T = +1.

2
2
2
2. Let T be a linear operator C which preserves the quadratic form x – x Show that
1 2
det T = 1.
32.6 Further Readings

Books Kenneth Hoffman and Ray Kunze, Linear Algebra
Michael, Artin Algebra

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